The theory of Sobolev extensions is of interest in several fields in analysis. Partial motivations for the study of Sobolev extensions comes from the thoery of PDEs, for example, see [1]}. It was proved in [2]}, [3]} that for every Lipschitz domain in \({{{\mathbb {R}}}^n}\) , there exists a bounded linear extension operator \(E\colon W^{k, p}({\Omega })\mapsto W^{k, p}({{{\mathbb {R}}}^n})\)  for each \(k\in \mathbb {N}\)  and \(1\le p\le \infty \) . Here \(W^{k, p}({\Omega })\)  is the Banach space of all \(L^p\) -integrable functions whose distributional derivatives up to order \(k\)  are \(L^p\) -integrable. Later, the notion of \((\epsilon , \delta )\) -domains was introduced by Jones in [4]}, and it was proved that for every \((\epsilon , \delta )\) -domain, there exists a bounded linear extension operator \(E\colon W^{k, p}({\Omega })\mapsto W^{k, p}({{{\mathbb {R}}}^n})\)  for every \(k\in \mathbb {N}\)  and \(1\le p\le \infty \) .
